To the numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials

Abstract

In the paper, computational schemes for solving the Cauchy problem for the singular integro-differential Prandtl equation with a singular integral over a segment of the real axis, understood in the sense of the Cauchy principal value, are constructed and justified. This equation is reduced to equivalent Fredholm equations of the second kind by inversion of the singular integral in three classes of Muskhelishvili functions and applying spectral relations for the singular integral. At the same time, we investigate the conditions for the solvability of integral Fredholm equations of the second kind with a logarithmic kernel of a special form and are approximately solved. The new computational schemes are based on applying the spectral relations for the singular integral to the integral entering into the equivalent equation. Uniform estimates of the errors of approximate solutions are obtained.